A geometric proof of the periodic averaging theorem on Riemannian manifolds

TL;DR

This paper provides a geometric proof of the averaging theorem for perturbed dynamical systems on Riemannian manifolds, extending previous results to cases with non-trivial $\

Contribution

It introduces a global averaging method on Riemannian manifolds that handles non-trivial $\

Findings

Established a geometric proof for the averaging theorem on Riemannian manifolds.

Extended averaging procedures to cases with non-trivial $\

Formulated results applicable on open domains with compact closure.

Abstract

We present a geometric proof of the averaging theorem for perturbed dynamical systems on a Riemannian manifold, in the case where the flow of the unperturbed vector field is periodic and the $\mathbb{S}^{1}$-action associated to this vector field is not necessarily trivial. We generalize the averaging procedure \cite{Arno-63,ArKN-88} defining a global averaging method based on a free coordinate approach which allow us to formulate our results on any open domain with compact closure.